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Real Part — Definition, Formula & Examples

Real Part

For a complex number a + bi, the real part is a.

 

Table showing complex numbers and their real parts: 6−2i→6, i−3→−3, 5i→0, 10+√2→10+√2

 

 

 

See also

Imaginary part

Key Formula

Re⁡(a+bi)=a\operatorname{Re}(a + bi) = a
Where:
  • aa = The real part of the complex number (a real number)
  • bb = The coefficient of the imaginary part (a real number)
  • ii = The imaginary unit, where i² = −1

Worked Example

Problem: Find the real part of the complex number 7 − 3i.
Step 1: Write the complex number in standard form a+bia + bi.
7−3i=7+(−3)i7 - 3i = 7 + (-3)i
Step 2: Identify aa, the term without ii.
a=7a = 7
Answer: The real part of 7−3i7 - 3i is 77, written Re⁡(7−3i)=7\operatorname{Re}(7 - 3i) = 7.

Why It Matters

Extracting the real part is essential when you need to separate a complex number into its two components for graphing on the complex plane, where the real part gives the horizontal coordinate. Many results in physics and engineering involve taking the real part of a complex expression to obtain a physically meaningful quantity, such as the actual voltage in an AC circuit.

Common Mistakes

Mistake: Confusing the real part with the imaginary part, especially when the number is written as a−bia - bi and mistakenly treating −b-b as the real part.
Correction: The real part is always the term that does not multiply ii. In a−bia - bi, the real part is aa and the imaginary part is −b-b.

Related Terms